In order to construct a proper \(\tilde{\mathcal {L}}^\varepsilon \) ,
using a diffeomorphic coordinate transformation \(y=\phi _s(x)\)  such that
\(\phi _s: \Omega _t\mapsto \Omega _{t+s}\)  (see [1]}),
the bijection \( V(\Omega _{t+s})\mapsto V(\Omega _{t})\) , \(u\mapsto u\circ \phi _s\) 
provides the perturbed Lagrangian as
\(\widetilde{\mathcal {L}}^\varepsilon (s, u\circ \phi _s, u\circ \phi _s, v\circ \phi _s; \Omega _t)=\mathcal {L}^\varepsilon (u, v; \Omega _{t+s}) \quad \text{for } (u, v)\in V(\Omega _{t+s})^2.\) 
